A rectangle is a four-sided shape, also known as a quadrilateral, where all four internal angles are right angles, measuring 90 degrees. It has several important properties that define it. The opposite sides of a rectangle are equal in length and parallel to each other.
Let's explore the key properties of a rectangle. First, all four angles are right angles, measuring 90 degrees. Second, opposite sides are equal in length - the two width sides are equal, and the two height sides are equal. Third, opposite sides are parallel to each other. Fourth, the diagonals bisect each other, meaning they cross at their midpoints. And fifth, both diagonals are equal in length, which is a special property of rectangles.
Now let's look at the important formulas related to rectangles. The perimeter of a rectangle equals two times the sum of its width and height, written as 2 times a plus b. For a rectangle with width 4 and height 2.5, the perimeter would be 2 times 6.5, which equals 13 units. The area of a rectangle is simply the product of its width and height, or a times b. In our example, that's 4 times 2.5, which equals 10 square units. The length of the diagonal can be calculated using the Pythagorean theorem, as the square root of a-squared plus b-squared. For our rectangle, that's approximately 4.7 units.
Let's explore some special types of rectangles. First, we have the square, which is a rectangle where all sides are equal in length. In a square, not only are all angles 90 degrees, but the diagonals are also equal in length and perpendicular to each other. This makes the square the most symmetrical of all rectangles. Another special type is the golden rectangle, where the ratio of width to height is equal to the golden ratio, approximately 1.618. The golden rectangle has been used throughout history in architecture and art because of its aesthetically pleasing proportions. Many famous buildings and artworks incorporate this special rectangle in their design.
To summarize what we've learned about rectangles: A rectangle is a four-sided shape or quadrilateral where all four internal angles are right angles, measuring 90 degrees. Opposite sides of a rectangle are equal in length and parallel to each other. The area of a rectangle is calculated by multiplying its width by its height, while the perimeter equals twice the sum of its width and height. The diagonals of a rectangle are equal in length and bisect each other at their midpoints. Special types of rectangles include the square, where all sides are equal, and the golden rectangle, which has proportions based on the golden ratio. Rectangles are fundamental shapes in geometry with numerous applications in mathematics, architecture, design, and everyday life.